7.3 The Thin-walled Pressure Vessel Theory

Section 7.3

7.3 The Thin-walled Pressure Vessel Theory

An important practical problem is that of a cylindrical or spherical object which is

subjected to an internal pressure p. Such a component is called a pressure vessel, Fig.

7.3.1. Applications arise in many areas, for example, the study of cellular organisms,

arteries, aerosol cans, scuba-diving tanks and right up to large-scale industrial containers

of liquids and gases.

In many applications it is valid to assume that

(i)

the material is isotropic

(ii)

the strains resulting from the pressures are small

(iii) the wall thickness t of the pressure vessel is much smaller than some

characteristic radius: t ? ro ? ri ?? ro , ri

p

t

2ri

2ro

Figure 7.3.1: A pressure vessel (cross-sectional view)

Because of (i,ii), the isotropic linear elastic model is used. Because of (iii), it will be

assumed that there is negligible variation in the stress field across the thickness of the

vessel, Fig. 7.3.2.

actual stress

?

p

?

t

?

p

approximate stress

?

t

Figure 7.3.2: Approximation to the stress arising in a pressure vessel

As a rule of thumb, if the thickness is less than a tenth of the vessel radius, then the actual

stress will vary by less than about 5% through the thickness, and in these cases the

constant stress assumption is valid.

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Section 7.3

Note that a pressure ? xx ? ? yy ? ? zz ? ? pi means that the stress on any plane drawn

inside the vessel is subjected to a normal stress ? pi and zero shear stress (see problem 6

in section 3.5.7).

7.3.1

Thin Walled Spheres

A thin-walled spherical shell is shown in Fig. 7.3.3. Because of the symmetry of the

sphere and of the pressure loading, the circumferential (or tangential or hoop) stress ? t

at any location and in any tangential orientation must be the same (and there will be zero

shear stresses).

?t

?t

Figure 7.3.3: a thin-walled spherical pressure vessel

Considering a free-body diagram of one half of the sphere, Fig. 7.3.4, force equilibrium

requires that

? ? ro2 ? ri 2 ? ? t ? ? ri 2 p ? 0

(7.3.1)

and so, with r0 ? ri ? t ,

ri 2 p

?t ?

2rti ? t 2

(7.3.2)

p

?t

Figure 7.3.4: a free body diagram of one half of the spherical pressure vessel

One can now take as a characteristic radius the dimension r. This could be the inner

radius, the outer radius, or the average of the two ¨C results for all three should be close.

Setting r ? ri and neglecting the small terms t 2 ? 2rti ,

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Section 7.3

?t ?

pr

2t

Tangential stress in a thin-walled spherical pressure vessel

(7.3.3)

This tangential stress accounts for the stress in the plane of the surface of the sphere. The

stress normal to the walls of the sphere is called the radial stress, ? r . The radial stress is

zero on the outer wall since that is a free surface. On the inner wall, the normal stress is

? r ? ? p , Fig. 7.3.5. From Eqn. 7.3.3, since t / r ?? 1 , p ?? ? t , and it is reasonable to

take ? r ? 0 not only on the outer wall, but on the inner wall also. The stress state in the

spherical wall is then one of plane stress.

?t

?t

?r ? 0

?r ? ?p ? 0

?t

Figure 7.3.5: An element at the surface of a spherical pressure vessel

There are no in-plane shear stresses in the spherical pressure vessel and so the tangential

and radial stresses are the principal stresses: ? 1 ? ? 2 ? ? t , and the minimum principal

stress is ? 3 ? ? r ? 0 . Thus the radial direction is one principal direction, and any two

perpendicular directions in the plane of the sphere¡¯s wall can be taken as the other two

principal directions.

Strain in the Thin-walled Sphere

The thin-walled pressure vessel expands when it is internally pressurised. This results in

three principal strains, the circumferential strain ? c (or tangential strain ? t ) in two

perpendicular in-plane directions, and the radial strain ? r . Referring to Fig. 7.3.6, these

strains are

?c ?

A?C ? ? AC C ?D ? ? CD

A?B ? ? AB

?

, ?r ?

AC

CD

AB

(7.3.4)

From Hooke¡¯s law (Eqns. 6.1.8 with z the radial direction, with ? r ? 0 ),

?? c ? ? 1 / E ? ? / E ? ? / E ? ?? t ?

?1 ? ? ?

?? ? ? ?? ? / E 1 / E ? ? / E ? ?? ? ? 1 pr ?1 ? ? ?

? c? ?

? ? t ? E 2t ?

?

??? r ?? ??? ? / E ? ? / E 1 / E ?? ??? r ??

??? 2? ??

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(7.3.5)

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Section 7.3

D?

D

C?

C

p

A

B

A? B?

before

after

Figure 7.3.6: Strain of an element at the surface of a spherical pressure vessel

To determine the amount by which the vessel expands, consider a circumference at

average radius r which moves out with a displacement ? r , Fig. 7.3.7. From the definition

of normal strain

?c ?

?r ? ? r ??? ? r?? ? ? r

r??

r

(7.3.6)

This is the circumferential strain for points on the mid-radius. The strain at other points

in the vessel can be approximated by this value.

The expansion of the sphere is thus

1 ?? pr 2

? r ? r? c ?

E 2t

(7.3.7)

?r ? ?r ???

?r

??

ro

ri

r

Figure 7.3.7: Deformation in the thin-walled sphere as it expands

To determine the amount by which the circumference increases in size, consider Fig.

7.3.8, which shows the original circumference at radius r of length c increase in size by

an amount ? c . One has

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Section 7.3

? c ? c? c ? 2?r? c ? 2?

1 ? ? pr 2

E 2t

(7.3.8)

It follows from Eqn. 7.3.7-8 that the circumference and radius increases are related

through

? c ? 2?? r

(7.3.9)

r

r ? ?r

c ? ? c ? 2? ?r ? ? r ?

c ? 2?r

Figure 7.3.8: Increase in circumference length as the vessel expands

Note that the circumferential strain is positive, since the circumference is increasing in

size, but the radial strain is negative since, as the vessel expands, the thickness decreases.

7.3.2

Thin Walled Cylinders

The analysis of a thin-walled internally-pressurised cylindrical vessel is similar to that of

the spherical vessel. The main difference is that the cylinder has three different principal

stress values, the circumferential stress, the radial stress, and the longitudinal stress ? l ,

which acts in the direction of the cylinder axis, Fig. 7.3.9.

p

?l

Figure 7.3.9: free body diagram of a cylindrical pressure vessel

Again taking a free-body diagram of the cylinder and carrying out an equilibrium

analysis, one finds that, as for the spherical vessel,

?l ?

pr

2t

Longitudinal stress in a thin-walled cylindrical pressure vessel (7.3.10)

Note that this analysis is only valid at positions sufficiently far away from the cylinder

ends, where it might be closed in by caps ¨C a more complex stress field would arise there.

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